Our present paper will be devoted to
a certain conceptual and methodical extension and completion
of the CHSP and PTQM approaches.
We felt motivated by the observation that
in a broader area of physics
both of these recipes
appeared comparatively
difficult to implement.
Various authors offered
remedies involving the
restriction of attention to the bounded-operator
Hamiltonians [1]} or the use of
specific models in which
the SP Hamiltonian \(H\)  is
\({\cal PT}-\) symmetric (see [2]}).
It is worth adding that
the latter approach was
based on a
parity-related (i.e., \({\cal P}-\) related)
and time-reversal-related (\({\cal T}-\) related) assumption
\(H{\cal PT}={\cal PT}H\) 
known as
\({\cal PT}-\) symmetry
of
the Hamiltonian.
This was an intuitively appealing feature
which made the PTQM approach particularly
popular, often even beyond its original scope and
restriction
to the closed and unitary quantum systems [3]}.
